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* Data from Larsen,1 in Petroleum Engineer, Nov. 1938

* Data from Larsen,1 in Petroleum Engineer, Nov. 1938

have a comparatively high ratio of water to solids, and the Qw/Qc ratio is correspondingly low. Table 6-2 compares the percent water retained in the cakes of three muds with the amount of water adsorbed by the dry clays in swelling tests. Note that the amount of water in the cakes is only slightly less than that in the swollen clays, and is virtually independent of the percentage of solids in the suspension. Indeed, the percent water in the cake is quite a good measure of the swelling properties of the clay base.

To a lesser extent, cake thickness is determined by particle size and particle-size distribution. These parameters control the porosity of the cake, and therefore the bulk volume relative to the grain volume. The magnitude of these effects was shown by Bo et al,9 who measured the porosities of filter cakes formed by mixing nine size grades of glass spheres. Their results may be summed up as follows:

1. Minimum porosities were obtained when there was an even gradation of particle sizes (i.e., a linear particle size distribution curve, as shown in Figures 6-5 a and b), because the smaller particles then packed most densely in the pores between the larger particles.

2. Mixtures with a wide range of particle sizes had lower porosities than mixtures with the same size distribution but narrower size range, (cf. Figure 6-5 a with b)

3. An excess of small particles resulted in lower porosities than did an excess of large particles.

We may expect the inert solids—which are comparatively isodimensional in shape—in drilling muds to exhibit similar phenomena. The behavior of the colloid fraction depends more on particle shape and on electrostatic forces, as discussed in the sections dealing with cake compressibility and cake permeability.

The thickness of the filter cake is difficult to measure accurately, largely because it is not possible to distinguish the boundary between the mud and the upper surface of the cake precisely. The problem arises because the cake is compacted by the hydraulic drag of the filtrate flowing through its pores. The hydraulic drag increases with depth below the surface of the cake, and the local pore pressure decreases correspondingly from the pressure of the mud on the surface of the cake to zero at the bottom of the cake. The compacting pressure (and the resulting intergranular stress) at any point is equal to the mud pressure less the pore pressure, and is therefore equal to zero in the surface layer, and to the mud pressure in the bottom layer of the cake. The distribution of intergranular stress and of density (expressed as porosity) with respect to distance from the bottom of the cake is shown in Figure 6-6 for theoretical and experimental values determined by Outmans4 with a suspension of ground calcium carbonate. Note that the distributions shown do not change with increase in thickness of the cake, so the average porosity of the cake remains constant with respect to time.

When accurate values of static cake thickness are required it is advisable to use the method developed by von Engelhardt and Schindewolf,3 which is as follows: only a limited amount of mud is put in the filtration cell, and filtration is stopped at the moment all of the mud is used up, so that only filter cake remains in the cell. The critical moment to stop filtration is determined by observing the filtrate volume at short time intervals, and concurrently plotting the volume versus the square root of the intervals. Filtration is stopped immediately when the curve departs from linearity. The total volume of mud filtered is calculated from the combined weight of the filtrate plus cake, divided by the density of the original mud. The cake volume is then obtained by subtracting the filtrate volume from the volume of mud filtered.

### The Permeability of the Filter Cake

The permeability of the filter cake is the fundamental parameter that controls both static and dynamic filtration. It more truly reflects downhole filtration behavior than does any other parameter. As a parameter for evaluating the filtration properties of muds with different concentration of

The above-investigators measured cake permeability from the filtration rate at the end of the test and from the cake thickness. Gates and Bowie1 -' mentioned the difficulty of measuring cake thickness accurately. This problem can be avoided by using the method of von Engelhardt and Schindewolfe.' described above, to determine cake volume, and then calculating permeabiluv from Equation 6-6, rearranged as follows:

When Qw and Qc are expressed in cm3, t in seconds, P in atmospheres, A in cm2, ft in centipoises, Equation 6-10 becomes then, with the standard API

test.

This method is suitable for laboratory studies of static filtration. At the wellsite, where accuracy is not so important, it is more convenient to measure the filter cake manually (see the last paragraph of this chapter), and to use Equation 6-6 in the form k =

If h is expressed in millimeters k = QwhfixS.95x 10"3 md

Note; with fresh water muds n is approximately one centipoise at 68 F (20 C). The Effect of Particle Size and Shape on Cake Permeability

Krumbein and Monk13 investigated the permeabilities of filter cakes of river sand by separating the sand into ten size fractions and recombining them to obtain two sets of mixtures. In one set, the mixtures had increasingly large mean particle diameters, but all had the same range of particle sizes, which were defined in terms of a parameter phi as shown in Figure 6-7. In the other set, all the mixtures had the same mean particle diameter, but increasingly wider ranges of particle sizes. The results showed that cake permeability decreased (1) with mean particle diameter, and (2) with increasing width of particles size range (see Figure 6-8).

One might expect minimum cake permeabilities with an even gradation of particle sizes. However, the experiments of Bo et al,9 already referred to, showed that minimum permeabilities were obtained when there was an excess r-r—I—r

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